Does Ignorance of the Whole Imply Ignorance of the Parts? Large Violations of Noncontextuality in Quantum Theory

A central question in our understanding of the physical world is how our knowledge of the whole relates to our knowledge of the individual parts. One aspect of this question is the following: to what extent does ignorance about a whole preclude knowledge of at least one of its parts? Relying purely on classical intuition, one would certainly be inclined to conjecture that a strong ignorance of the whole cannot come without significant ignorance of at least one of its parts. Indeed, we show that this reasoning holds in any noncontextual (NC) hidden-variable model (HV). Curiously, however, such a conjecture is false in quantum theory: we provide an explicit example where a large ignorance about the whole can coexist with an almost perfect knowledge of each of its parts. More specifically, we provide a simple information-theoretic inequality satisfied in any NC HV, but which can be arbitrarily violated by quantum mechanics.

Figure 1

One can understand our result in terms of a game between Bob and a malicious challenger, the Owl. Imagine Bob is taking a philosophy class teaching him knowledge about $Y$, chosen uniformly at random. Unfortunately, he never actually attended and had insufficient time to prepare for his exam. Luckily, however, he has been given an encoding $E$ of the possible answers $Y_0{Y}_1$, hastily prepared by his old friend Alice. When entering the room, he had to submit $E$ for inspection to the challenger who knows $Y_0$, $Y_1$ as well as the encoding Alice might use. After inspection, the challenger may secretly keep a system $C$, possibly correlated with $E$, but such that the reduced system on $Y_0$, $Y_1$ and $E$ looks untampered with. It is immediately obvious to the challenger that Bob must be ignorant about the whole of $Y_0{Y}_1$. But can it always measure and point to a $C=c$ such that Bob is ignorant about $Y_c$? Classically, this is indeed possible: ignorance about the whole of $Y_0{Y}_1$ implies significant ignorance about one of the parts, $Y_C$. However, a quantum Bob could beat the Owl.